This is a geometric sequence with explicit function:

DEFINITION Limit of a Sequence We say that {an} converges to a limit L, and we write if, for every ε > 0, there is a number M such that |an − L| < ε for all n > M. If no limit exists, we say that {an} diverges. If the terms increase without bound, we say that {an} diverges to infinity.

If {an} converges, then its limit L is unique If {an} converges, then its limit L is unique. A good way to visualize the limit is to plot the points (1, a1), (2, a2), (3, a3),…. The sequence converges to L if, for every ε > 0, the plotted points eventually remain within an ε-band around the horizontal line y = L.

The figure below shows the plot of a sequence converging to L = 1.

On the other hand, we can show that the sequence an = cos n has no limit.

Solution The definition requires us to find, for every ε > 0, a number M such that

Note the following two facts about sequences: The limit does not change if we change or drop finitely many terms of the sequence. If C is a constant and an = C for all n sufficiently large, then Many of the sequences we consider are defined by functions; that is, an = f (n) for some function f (x). For example,

If f (x) converges to L, then the sequence an = f (n) also converges to L.

Find the limit of the sequence THEOREM 1 Sequence Defined by a Function Find the limit of the sequence

THEOREM 1 Sequence Defined by a Function

If r > 1, the geometric sequence an = crn diverges to ∞. A geometric sequence is a sequence an = crn, where c and r are nonzero constants. Each term is r times the previous term; that is, an/an-1 = r. The number r is called the common ratio. For instance, if r = 3 and c = 2, we obtain the sequence (starting at n = 0) In the next example, we determine when a geometric sequence converges. Recall that {an} diverges to ∞ if the terms an increase beyond all bounds. If r > 1, the geometric sequence an = crn diverges to ∞.

Geometric Sequences with r ≥ 0 Prove that for r ≥ 0 and c > 0, f (x) = crx. If 0 ≤ r < 1, then If r > 1, then an= crn diverges to ∞ (because c > 0). If r = 1, then crn = c for all n, and the limit is c.

THEOREM 2 Limit Laws for Sequences Assume that {an} and {bn} are convergent sequences with Then:

THEOREM 3 Squeeze Theorem for Sequences Let {an}, {bn}, {cn} be sequences such that for some number M,

Geometric Sequences with r < 0 Prove that for c ` If -1 < r < 0, then If r = -1, then If r < -1, then

As another application of the Squeeze Theorem, consider the sequence

THEOREM 4 If f (x) is continuous and In other words, we may “bring a limit inside a continuous function.”

Of great importance for understanding convergence are the concepts of a bounded sequence and a monotonic sequence. DEFINITION Bounded Sequences A sequence {an} is: Bounded from above if there is a number M such that an ≤ M for all n. The number M is called an upper bound. Bounded from below if there is a number m such that an ≥ m for all n. The number m is called a lower bound. The sequence {an} is called bounded if it is bounded from above and below. A sequence that is not bounded is called an unbounded sequence.

Upper and lower bounds are not unique Upper and lower bounds are not unique. If M is an upper bound, then any larger number is also an upper bound, and if m is a lower bound, then any smaller number is also a lower bound.

As we might expect, a convergent sequence {an} is necessarily bounded because the terms an get closer and closer to the limit. This fact is recorded in the next theorem. THEOREM 5 Convergent Sequences Are Bounded If {an} converges, then {an} is bounded.

There are two ways that a sequence {an} can diverge There are two ways that a sequence {an} can diverge. One way is by being unbounded. For example, the unbounded sequence an = n diverges: 1, 2, 3, 4, 5, 6,… However, a sequence can diverge even if it is bounded. This is the case with an = (−1)n+1, whose terms an bounce back and forth but never settle down to approach a limit: 1, −1, 1, −1, 1, −1,… There is no surefire method for determining whether a sequence {an} converges, unless the sequence happens to be both bounded and monotonic. By definition, {an} is monotonic if it is either increasing or decreasing:

Intuitively, if {an} is increasing and bounded above by M, then the terms must bunch up near some limiting value L that is not greater than M.

THEOREM 6 Bounded Monotonic Sequences Converge If {an} is increasing and an ≤ M, then {an} converges and If {an} is decreasing and an ≥ m, then {an} converges and